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Suppose there is a object M, (sliding motion) moving by the initial speed $v$ and the initial location $x_0$. Otherwise noted, friction is assumed to be nonexistent. It then meets a circular mold with the radius $R$ that the object travels (perimeter). After traveling one round of the mold, it moves onto the right straight path then meets with the path with friction, given by friction coefficient $\mu_k$. When colliding with the wall at $x=d$, the collision is perfectly elastic collision. The object moves back to the original initial position, $x_0$, then bounces with some object, with perfect elastic collision, back to continue the cycle.

the movement of the object

The question is, I do know how to describe the movement of the object in Newtonian way, but I would like to know how I would be able to describe the movement using Lagrangian mechanics (or Hamiltonian mechanics).

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Having friction in the problem, it might not be easy to come up with a Hamiltonian/Lagrangian description. How desperately you want such a description and how much you are willing to suffer? –  Fabian Oct 18 '12 at 9:18
Is there a lagrangian that describes a much simpler problem i.e body moving under the Influence of friction. If that is know you can generalize it. –  Prathyush Oct 18 '12 at 10:11
@Prathyush No... The very reason I asked this question is that Lagrangian way would be somehow difficult, but I needed it as somehow smaller part of my assignment. (The assignment obviously wasn't this.) –  War Oct 18 '12 at 10:26
I am willing to suffer much. It's OK to use any mathematical techniques (I guess this won't be the abstract algebra stuff, right?). –  War Oct 18 '12 at 10:27
what does google say? –  Prathyush Oct 18 '12 at 10:49

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